Triangle calculator SSA

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Triangle has two solutions with side c=65.35223498236 and with side c=3.25992554143

#1 Obtuse scalene triangle.

Sides: a = 37   b = 34   c = 65.35223498236

Area: T = 452.9066291074
Perimeter: p = 136.3522349824
Semiperimeter: s = 68.17661749118

Angle ∠ A = α = 24.05879314829° = 24°3'29″ = 0.42198901156 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 133.9422068517° = 133°56'31″ = 2.33877301026 rad

Height: ha = 24.48114211392
Height: hb = 26.64215465338
Height: hc = 13.86604439564

Median: ma = 48.69551210465
Median: mb = 50.309869521
Median: mc = 13.95659160621

Inradius: r = 6.64331754445
Circumradius: R = 45.38109417634

Vertex coordinates: A[65.35223498236; 0] B[0; 0] C[34.3065802619; 13.86604439564]
Centroid: CG[33.21993841475; 4.62201479855]
Coordinates of the circumscribed circle: U[32.67661749118; -31.49112284369]
Coordinates of the inscribed circle: I[34.17661749118; 6.64331754445]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.9422068517° = 155°56'31″ = 0.42198901156 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 46.05879314829° = 46°3'29″ = 2.33877301026 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 37 ; ; b = 34 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 37**2 + c**2 -2 * 37 * c * cos (22° ) ; ; ; ; c**2 -68.612c +213 =0 ; ; p=1; q=-68.612; r=213 ; ; D = q**2 - 4pr = 68.612**2 - 4 * 1 * 213 = 3855.55237333 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 68.61 ± sqrt{ 3855.55 } }{ 2 } ; ;
c_{1,2} = 34.30580262 ± 31.0465472047 ; ; c_{1} = 65.3523498236 ; ; c_{2} = 3.2592554143 ; ; ; ; text{ Factored form: } ; ; (c -65.3523498236) (c -3.2592554143) = 0 ; ; ; ; c>0 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 37 ; ; b = 34 ; ; c = 65.35 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 37+34+65.35 = 136.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 136.35 }{ 2 } = 68.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 68.18 * (68.18-37)(68.18-34)(68.18-65.35) } ; ; T = sqrt{ 205124.11 } = 452.91 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 452.91 }{ 37 } = 24.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 452.91 }{ 34 } = 26.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 452.91 }{ 65.35 } = 13.86 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 34**2+65.35**2-37**2 }{ 2 * 34 * 65.35 } ) = 24° 3'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37**2+65.35**2-34**2 }{ 2 * 37 * 65.35 } ) = 22° ; ;
 gamma = 180° - alpha - beta = 180° - 24° 3'29" - 22° = 133° 56'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 452.91 }{ 68.18 } = 6.64 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37 }{ 2 * sin 24° 3'29" } = 45.38 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 65.35**2 - 37**2 } }{ 2 } = 48.695 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 65.35**2+2 * 37**2 - 34**2 } }{ 2 } = 50.309 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 37**2 - 65.35**2 } }{ 2 } = 13.956 ; ;



#2 Obtuse scalene triangle.

Sides: a = 37   b = 34   c = 3.25992554143

Area: T = 22.58773635047
Perimeter: p = 74.25992554143
Semiperimeter: s = 37.13296277072

Angle ∠ A = α = 155.9422068517° = 155°56'31″ = 2.7221702538 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 2.05879314829° = 2°3'29″ = 0.03659176802 rad

Height: ha = 1.22109385678
Height: hb = 1.32986684415
Height: hc = 13.86604439564

Median: ma = 15.52661512593
Median: mb = 20.02202740473
Median: mc = 35.49442856462

Inradius: r = 0.60883380012
Circumradius: R = 45.38109417634

Vertex coordinates: A[3.25992554143; 0] B[0; 0] C[34.3065802619; 13.86604439564]
Centroid: CG[12.52216860111; 4.62201479855]
Coordinates of the circumscribed circle: U[1.63296277072; 45.35216723933]
Coordinates of the inscribed circle: I[3.13296277072; 0.60883380012]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 24.05879314829° = 24°3'29″ = 2.7221702538 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 177.9422068517° = 177°56'31″ = 0.03659176802 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 37 ; ; b = 34 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 37**2 + c**2 -2 * 37 * c * cos (22° ) ; ; ; ; c**2 -68.612c +213 =0 ; ; p=1; q=-68.612; r=213 ; ; D = q**2 - 4pr = 68.612**2 - 4 * 1 * 213 = 3855.55237333 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 68.61 ± sqrt{ 3855.55 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 34.30580262 ± 31.0465472047 ; ; c_{1} = 65.3523498236 ; ; c_{2} = 3.2592554143 ; ; ; ; text{ Factored form: } ; ; (c -65.3523498236) (c -3.2592554143) = 0 ; ; ; ; c>0 ; ; : Nr. 1
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 37 ; ; b = 34 ; ; c = 3.26 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 37+34+3.26 = 74.26 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.26 }{ 2 } = 37.13 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.13 * (37.13-37)(37.13-34)(37.13-3.26) } ; ; T = sqrt{ 510.19 } = 22.59 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.59 }{ 37 } = 1.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.59 }{ 34 } = 1.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.59 }{ 3.26 } = 13.86 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 34**2+3.26**2-37**2 }{ 2 * 34 * 3.26 } ) = 155° 56'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37**2+3.26**2-34**2 }{ 2 * 37 * 3.26 } ) = 22° ; ;
 gamma = 180° - alpha - beta = 180° - 155° 56'31" - 22° = 2° 3'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.59 }{ 37.13 } = 0.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37 }{ 2 * sin 155° 56'31" } = 45.38 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 3.26**2 - 37**2 } }{ 2 } = 15.526 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.26**2+2 * 37**2 - 34**2 } }{ 2 } = 20.02 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 37**2 - 3.26**2 } }{ 2 } = 35.494 ; ;
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